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lieven lebruyn
Works at University of Antwerp
Attended Universiteit Antwerpen
Lives in antwerp,belgium
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crowd-funding Grothendiecks biography? #Grothendieck  

+John Baez has a post out at the n-cat-cafe on Leila Schneps's quest to raise $6000 to translate Scharlau's 3-volume biography of Grothendieck http://bit.ly/1mfKpca .

If you care to contribute : http://www.gofundme.com/7ldiwo

Lots of good stuff in volume 3 on Groths hippy/eco/weirdo years. I've plundered Scharlau's text last year trying to pinpoint the location of Groths hideout in the French Pyrenees (http://bit.ly/1dlV6BN).

As far as i know, part 2 (the most interesting part on Groths mathematical years) is still under construction and will be compiled by the jolly group called the "Grothendieck circle" http://bit.ly/PMvnxA.

There's a nice series of G-recollections out here (http://bit.ly/1hxgt5u) (a.o. by Illusie, Karoubi, Cartier, Raynaud, Mumford, Hartshorne, Murre, Oort, Manin, Cartier).

I'm pretty sure Groth himself would prefer we'd try to get his Recoltes et Semailles (http://bit.ly/Nrh9QG) translated into English, or La Clef des Songes (http://bit.ly/1fFA01i)
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He was apparently very fond of it, though I seem to remember reading that at the time, it seemed to read as *here is a list of all the things I don't like about people I used to work with*, which may not have helped the books' popularity much.
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Why did Grothendieck quit mathematics?

After yesterday's post on the striking similarities between the lives of Grothendieck and JD Salinger http://goo.gl/v1QeyP it sure felt weird to stumble upon this footnote in "La Clef des Songes"  http://goo.gl/AcPC5A.
Probably I'm reading way too much into it, but it appears to indicate that Grothendieck stopped doing mathematics to become ... a writer!

#Grothendieck  
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I didn't know that! Maybe you're correct, then. Récoltes et Semailles could easily have been called "Things I really need to get off my chest" though. In a way, it is a significant litterary achievement, since I've read it and haven't read "Pursuing Stacks" for example.
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apologies to fellow M-club members for lack of M-posts recently. i'm teaching 2 bachelor and 2 master courses this semester. probably should've decided to lecture on Mochizuki's papers in one of the master courses, but then, i didn't...   #Mochizuki #ABC
 
Mochizuki club circle 
Here's everyone who showed interest (posted) when Shinichi Mochizuki announced he had proof for the abc- conjecture. It's been great fun to read the impressions from back then.

I have seen especially interesting posts by +John Baez, who gave very accessible first fragmentary impressions. Just search for his name and "mochizuki" and don't forget to read the comments!

Also, i urge you to look at +lieven lebruyn's profile, who may be the one who got deepest into explaining it. And look: Colored, hand drawn diagrams! He's been posting MinuteMochizuki on his blog until mid of june and i think he'll be back!

http://matrix.cmi.ua.ac.be/ 

For more here's the polymath page which, among many other interesting links, lists quite a reaction here on google+. +Jordan Ellenberg +Timothy Gowers +Terence Tao  also made it into this circle.

http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture

As for where matters stand, it may be that Mochizuki is currently giving private lectures to Go Yamashita, a colleague, who might be inclined to do the travelling to explain it to others. Or so... that's what this blog post by Caroline Chen suggests:

http://projectwordsworth.com/the-paradox-of-the-proof/

Minhyong Kim, a friend of Mochizuki at Oxford, has been posting on mathoverflow an early attempt to give some overview from a closer vantage point:

http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture

And, as i must say, i stole the following taglist from +Rusty Ferguson's latest circle share. You wrote some appealing stuff, your tags must be good:
https://plus.google.com/112078832938393562132/posts/WvqYjBrKiir

#sharedcircles   #publicsharedcircles   #circleshare   #sharedpublicricle   #circlesharing   #circle   #circleoftheday   #circleoftheweek  
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what i'll be doing the coming weeks:

point as many square metres exterior walls as possible of an old vernacular building.  huge cracks (center) in urgent need of repair.
fortunately these are not structural problems as the walls are close to 1 metre thick, but still it's better to have the job done before another rainy autumn and freezing winter...
so wish me: not too much rain, not too much wind and a miraculous cure from vertigo...
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I wish you not too much rain, not too much wind, a miraculous cure from vertigo, a huge amount of energy and a nice rosé during the evenings.
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lieven lebruyn

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Mochizuki's Frobenioid reconstruction: the final bit

In "The geometry of Frobenioids 1" http://goo.gl/3tWTC Mochizuki 'dismantles' arithmetic schemes and replaces them by huge categories called Frobenioids. Clearly, one then wants to reconstruct the schemes from these categories and we almost understood how he manages to to this ( see http://goo.gl/TbMLl) modulo the 'problem' that there might be auto-equivalences of Frob(Z), the Frobenioid corresponding to Spec(Z) i.e.the collection of all prime numbers, reshuffling distinct prime numbers.

In previous posts i've simplified things a lot, leaving out the 'Arakelov' information contained in the infinite primes, and feared that this lost info might be crucial to understand the final bit. Mochizuki's email http://goo.gl/42dd3 also points in that direction.

Here's what i hope to have learned this week:

the full Frobenioid Frob(Z)

objects of Frob(Z) consist of pairs (q,r) where q is a strictly positive rational number and r is a real number.

morphisms are of the form f=(n,a,(z,s)) : (q,r) --> (q',r') (where n and z are strictly positive integers, a a strictly positive rational number and s a positive real number) subject to the relations that

q^n.z = q'.a  and n.r+s = r'+log(a)

n is called the Frobenious degree of f and (z,s) the divisor of f.

All this may look horribly complicated until you realise that the isomorphism classes in Frob(Z) are exactly the 'curves' C(alpha) consisting of all pairs (q,r) such that r-log(q)=alpha, and that morphisms with the same parameters as f send points in C(alpha) to points in C(beta) where

beta = n.alpha + s - log(a)

So, the isomorphism classes can be identified with the real numbers R and special linear morphisms of type (1,a,(1,s)) and their compositions are compatible with the order-structure and addition on R.

Crucial is Mochizuki's observation that C(0) are precisely the 'Frobenious-trivial' objects in Frob(Z) (i'll spare you the details but it is a property on having sufficiently many nice endomorphisms).

Now, consider an auto-equivalence E of Frob(Z). It will induce a map between the isoclasses E : R --> R which is additive and as Frob-trivs are mapped under E to Frob-trivs this will map 0 to 0, so E will be an additive group-endomorphism on R hence of the form x --> r.x for some fixed real number r, and we want to show that r=1.

Linear irreducible morphisms are of type (1,a,(p,0)) where p is a prime number and they map C(alpha) to C(alpha-log(p)). As irreducibles are preserved under equivalence this means that for each prime p there must exist a prime q such that r log(p) = log(q)

Now if r is an irrational number, there must be at least three triples (p1,q1),(p2,q2) and (p3,q3) satisfying r log(pi) = log(qi) but this contradicts a fairly hard result due to Lang that for 6 distinct primes l1,...,l6 there do not exist positive rational numbers a,b such that

log(l1)/log(l2) = a log(l3)/log(l4) = b log(l5)/log(l6)

So, r must be rational and of the form n/m, but then r=1 (if not the correspondence r.log(p) = log(q) gives p^n=q^m contradicting unique factorisation). This then shows that under the auto-equivalence each prime p (corresponding to a linear irreducible map) is send to itself.

This was the remaining bit left to show (as in http://goo.gl/TbMLl) that the Frobenioid corresponding to any Galois extension of the rationals contains enough information to reconstruct from it the schemes of all rings of integers in intermediate fields.
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Very good!
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lieven lebruyn

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while on the topic of inspirational math movies, here's my all time favourite

about 10 years ago i planned to start every fresh(wo)men course  watching  'good will hunting' for 2 hours instead of teaching, if they'd promise me to write half a page, pinpointing the scene they'd figure was my prime reason for wasting all this time...

perhaps fortunately i didn't pursue the idea (are you familiar with teaching evaluations?).

anyway, here's the full marks answer (only for those who watched the movie at least once) .
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ahh ok sorry, i had misunderstood the "wasting time" part... this is of course out of pure personal bias (my own description of what i do all day).
i remember i saw the pic of your house on g+, nice job! 
what i do love about the clip, and the movie, is the display of friendship, which is probably what you meant in the first place. 
in some sense, regarding my over-zealous reply, i have to admit in hindsight that there is also nothing wrong with doing these jobs out of some sense of duty to the advancement of humanity, no matter how tiny our own epsilon contributions may be... more than that, maybe we all SHOULD feel this sense of duty also, along with our other motivations. if we ourselves don't endorse the public worth of what we do over some of the other endeavours (say, oh i don't know, banking ;)), out of a sense of false modesty, that is also irresponsible. 
so yeah, think first, type later ;)
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Grothendieck and J.D. Salinger

It's hardly original to call Grothendieck the Pynchon or Salinger of mathematics (read for example Sam Leith's 'the Einstein of maths' http://goo.gl/Ddk79m), yet the biography  of J.D. (http://goo.gl/RpV8Ik, a new-year's present from my father) almost reads like a copy/paste of A.G.'s life...

- both were damaged by WW2 experiences
- both opted out at the height of their success 
- both kept on writing in seclusion
- both are said to have left a secret treasure
- both left a trail of failed relationships (wives, lovers and children)
- both turned to religion
- both had dietary problems
- etc. etc.

Try to substitute 'Grothendieck' for 'Salinger' (and 'EGA' for 'Catcher in the Rye') in this trailer of the documentary film on which David Shields and Shane Salerno's book is based...

#Grothendieck
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lieven lebruyn

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for Grothendieck aficionados
a chance discovery last month en route from Les Vans - Lablachere (in the Ardeche region), a 'ferronnerie d'art' (a wrought-iron workshop) called 'La Clef des Songes'.
All 315 pages of this Grothendieck meditation from 1987 can be found at http://goo.gl/AcPC5A
The 691 pages of 'Notes pour la clef des songes' are a bit harder to get. Fortunately, the mysterious website 'l'astree' offers them as a series of 23 pdfs http://goo.gl/Oy0qvC Enjoy the read!

I don't understand why this post continues to disappear. As I'd like it to remain on my profile-posts page this edit.
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Thanks for finding that, +lieven lebruyn 
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Szpiro's Marabout-Flash on number theory

For travellers into Mochizuki-territory the indispensable rough guide is Lucien Szpiro's 'Marabout-Flash de théorie des nombres algébriques', aka section I.1.3 in  'Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell' http://goo.gl/6npMg7.

Marabout Flash http://goo.gl/T5EIzV was a Franco-Belge series of do-it-yourself booklets, quite popular in the 60ties and 70ties, on almost every aspect of everyday's life.

In just a couple of pages Szpiro describes how one can extend the structure sheaf of a fractional ideal of a ring of integers to a 'metrized' (or Arakelov) line-bundle on the completed prime spectrum (including the infinite places). These bundles then satisfy properties similar to those of line-bundles on smooth projective curves, including a version of the Riemann-Roch theorem and a criterium to have non-zero global sections.

These (fairly simple) results then quickly lead to proofs of the first major results in number theory such as the Hermite-Minkowski theorem and Dirichlet's unit theorem.

#minutemochizuki  
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On a side note, I like the typesetting, which shows that pre-LaTeX monospace doesn't have to be ugly or illegible. (There's a book of A. Robert on representation theory and locally compact groups typeset in the same way)
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lieven lebruyn

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Mochizuki update

While i've been away from G+, someone emailed Mochizuki my last post http://goo.gl/0LKQO on the problem i have with Frobenioids1. He was kind enough to forward  me M's reply:

"There is absolutely nothing difficult or subtle going on here (e.g., by comparison to the portion of the theory cited in the discussion preceding the statement of this ``problem''!).  The nontrivial result is the fact that the degree is rational, i.e., the initial portion of [FrdI], Theorem 6.4, (iii), which is a consequence of a highly nontrivial result in transcendence theory due to Lang (i.e., [FrdI], Lemma 6.5, (ii)).  Once one knows this rationality, the conclusion that distinct prime numbers are not confused with one another is a formal consequence (i.e., no complicated subtle arguments!) of the fact that the ratio of the natural logarithm of any two distinct prime numbers is never rational.
Sincerely, Shinichi Mochizuki"

Being back, i'll give it another go.

Having read the shared article below, it's comforting to know that other people, including  +Aise Johan de Jong and +Cathy O'Neil, are also frustrated by M's opaque latest writings.
 
On Shinichi Mochizuki's proposed proof of the ABC conjecture:

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The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”

This is not just gibberish to the average layman. It was gibberish to the math community as well.
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I think that Thurston's article On proof and progress in mathematics should be mandatory reading for all  math undergraduate students.
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Have him in circles
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mathematician by day, blogger by night
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early adopter of noncommutative geometry, sceptic evangelist of the 'field with one element', passionate about a 400 year old chestnut farm somewhere in the Cevennes.
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